On Geodesically Convex Formulations for the Brascamp-Lieb Constant
On Geodesically Convex Formulations for the Brascamp-Lieb Constant
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关于 Brascamp-Lieb 常数的测地凸公式
DOI：
10.4230/lipics.approx-random.2018.25
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发表时间：
2018
影响因子：
3.1
通讯作者：
Ozan Yildiz
中科院分区：
文献类型：
--
作者：
Nisheeth K. Vishnoi;Ozan Yildiz
我们考虑了对应于给定基准的两种计算Brascamp-Lieb不等式中最优常数的非凸公式，并证明了它们在具有对应于对数行列式函数的Hessian的黎曼度量的正定矩阵流形上是测地对数凹的。第一种形式出现在Lieb的工作中，第二种形式受到Bennett等人的工作的启发。Garg et al.和Allen-Zhu et al.最近的作品也通过对算子尺度问题的简化暗示了Brascamp-Lieb常数的测地线对数凹公式。然而，在它们的缩减中出现的优化问题的维度指数地依赖于描述Brascamp-Lieb基准所需的比特数。这里给出的公式的维度是输入数据的位复杂度的多项式。
We consider two non-convex formulations for computing the optimal constant in the Brascamp-Lieb inequality corresponding to a given datum, and show that they are geodesically log-concave on the manifold of positive definite matrices endowed with the Riemannian metric corresponding to the Hessian of the log-determinant function. The first formulation is present in the work of Lieb and the second is inspired by the work of Bennett et al. Recent works of Garg et al.and Allen-Zhu et al. also imply a geodesically log-concave formulation of the Brascamp-Lieb constant through a reduction to the operator scaling problem. However, the dimension of the arising optimization problem in their reduction depends exponentially on the number of bits needed to describe the Brascamp-Lieb datum. The formulations presented here have dimensions that are polynomial in the bit complexity of the input datum.